I am working out of Miranda's Algebraic Curves and Riemann Surfaces, and I am unsure of how to proceed on problem III.2.A.
The problem is this: Let $X$ be the smooth affine plane curve defined by the equation $y^2=h(x)$ where $h(x)$ is a polynomial of degree $2g+1$ with distinct roots. Show that the map $\phi(x,y) = y/x^{g+1}$ defines a hole chart on $X$ for $|x|$ large.
Intuitively, this is clear to me. As we let $|x|$ grow large, $\phi(x,y)$ will grow small, but it never reaches 0. This is where the hole is. My question is that I do not know how to clearly show that $\phi$ is even a regular chart to begin with. Once I can do that I feel like the rest of the problem is not so bad.
Here is what I have written so far: Let $M$ be the maximum of the absolute values of all of the roots of $h(x)$ and $h'(x)$. Write $K = \{(x,y) ; |x|>M \}$. This set $K$ is open. Then for $(x,y)\in K$ it follows that $\frac{\partial f}{\partial y}$ and $\frac{\partial f}{\partial x}$ are non-zero by our requirement on $x$.
Then somehow I would like to whisk these facts together to show that $\phi(x,y)$ is a homeomorphism of $K$ with $\phi(K)$. This is where I get lost. I tried letting $y=y(x)$. Assuming that $y(x)$ is equal to the positive branch of $\sqrt{h(x)}$, then I can write $$ \phi(x) = \sqrt{h(x)}/x^{g+1}. $$ Maybe from here I can use the invertible function theorem? I am not sure though, and I feel like I'm getting into the weeds a bit.